The concepts of Mandelbrot's fractal geometry have been applied to the structure of individual central nervous system neurons and other cell types grown in tissue culture or from whole animals. Techniques have been developed to measure the "fractal dimension" (FD), which is a measure of complexity of individual cells' structure, with particular reference to the degree of their dendritic branching and the roughness of their borders. These techniques were calibrated against images of known FD and then employed to measure the unknown FD of individual neurons. The range of FD's in 28 neurons examined was between 1.14, indicating a relatively low complexity, to 1.60, indicating a high complexity. In addition, we have applied our methods to other cell types (e.g., cat cortical pyramidal neurons, glia, etc.) Our research on cat cortical pyramidal neurons has shown that different pyramidal neuron types from different areas of the motor cortex have different fractal dimensions, with the largest found in those cells that make up the pyramidal track. Our work on glia cells has shown that primoidal cells that mature into oligodendrocytes develop larger fractal dimensions and at a faster rate than astrocytes. The most remarkable finding, however, is that the rate of development of the fractal dimension with time can be described remarkably well by a single time constant.